DISCUSSION PAPER SERIES
DP12085
EMPIRICAL EVALUATION OF
OVERSPECIFIED ASSET PRICING
MODELS
Elena Manresa, Francisco Peñaranda and Enrique
Sentana
ISSN 0265-8003
EMPIRICAL EVALUATION OF OVERSPECIFIED
ASSET PRICING MODELS
Elena Manresa, Francisco Peñaranda and Enrique Sentana
Discussion Paper DP12085
Published 11 June 2017
Submitted 11 June 2017
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EMPIRICAL EVALUATION OF OVERSPECIFIED
ASSET PRICING MODELS
Abstract
Asset pricing models with potentially too many risk factors are increasingly common in empirical
work. Unfortunately, they can yield misleading statistical inferences. Unlike other studies
focusing on the properties of standard estimators and tests, we estimate the sets of SDFs and
risk prices compatible with the asset pricing restrictions of a given model. We also propose tests
to detect problematic situations with economically meaningless SDFs uncorrelated to the test
assets. We confirm the empirical relevance of our proposed estimators and tests with Yogo's
(2006) linearized version of the consumption CAPM, and provide Monte Carlo evidence on their
reliability in finite samples.
JEL Classification: G12, C52
Keywords: Continuously Updated GMM, Factor pricing models, Set estimation, Stochastic
discount factor, Underidentification tests.
Elena Manresa - [email protected]
MIT Sloan
Francisco Peñaranda - [email protected]
Queens College CUNY
Enrique Sentana - [email protected]
CEMFI and CEPR
Acknowledgements
We would like to thank Craig Burnside and Motohiro Yogo for kindly making their data available to us. We also thank Cesare Robotti, Enrique Salvador, participants at the Finance Forum (Zaragoza), the Midwest Finance Association Meeting (Chicago), the UK Money and Macro Finance Research Group Workshop on "Empirical Modelling of Financial Markets" (Brunel), the Annual SoFiE Conference (Aarhus), the International Conference on Computational and Financial Econometrics (Sevilla) and the Econometric Society World Congress (Montreal), as well as audiences at CEU Cardenal Herrera (Elche), Fundação Getulio Vargas (Rio), Geneva, Harvard Business School, LSE, McCombs School of Business (Austin) and Queens College CUNY for helpful comments and suggestions. Of course, the usual caveat applies. Financial support from the Spanish Government through grants ECO 2011-25607 (Peñaranda), 2011-26342 and 2014-59262 (Sentana) is gratefully acknowledged. The second author is also very grateful to SanFI for its funding.
Empirical Evaluation of
Overspeci…ed Asset Pricing Models
Elena Manresa
MIT Sloan, 30 Memorial Dr, Cambridge, MA 02142, US <[email protected]>
Francisco Peñaranda
Queens College CUNY, 65-30 Kissena Blvd, Flushing, NY 11367, US <[email protected]>
Enrique Sentana
CEMFI, Casado del Alisal 5, E-28014 Madrid, Spain <sentana@cem….es>
June 2014. Revised: May 2017
Abstract
Asset pricing models with potentially too many risk factors are increasingly common in empirical work. Unfortunately, they can yield misleading statistical inferences. Unlike other studies focusing on the properties of standard estimators and tests, we estimate the sets of SDFs and risk prices compatible with the asset pricing restrictions of a given model. We also propose tests to detect problematic situations with economically meaningless SDFs uncorrelated to the test assets. We con…rm the empirical relevance of our proposed estimators and tests with Yogo’s (2006) linearized version of the consumption CAPM, and provide Monte Carlo evidence on their reliability in …nite samples.
Keywords: Continuously Updated GMM, Factor pricing models, Set estimation, Stochastic discount factor, Underidenti…cation tests.
JEL: G12, C52.
We would like to thank Craig Burnside and Motohiro Yogo for kindly making their data available to us. We also thank Cesare Robotti, Enrique Salvador, participants at the Finance Forum (Zaragoza), the Midwest Finance Association Meeting (Chicago), the UK Money and Macro Finance Research Group Workshop on “Empirical Mod-elling of Financial Markets” (Brunel), the Annual SoFiE Conference (Aarhus), the International Conference on Computational and Financial Econometrics (Sevilla) and the Econometric Society World Congress (Montreal), as well as audiences at CEU Cardenal Herrera (Elche), Fundação Getulio Vargas (Rio), Geneva, Harvard Business School, LSE, McCombs School of Business (Austin) and Queens College CUNY for helpful comments and sugges-tions. Of course, the usual caveat applies. Financial support from the Spanish Government through grants ECO 2011-25607 (Peñaranda), 2011-26342 and 2014-59262 (Sentana) is gratefully acknowledged. The second author is also very grateful to SanFI for its funding.
1
Introduction
The most popular empirically oriented asset pricing models e¤ectively assume the existence of a common stochastic discount factor (SDF) that is linear in some risk factors, which discounts uncertain payo¤s di¤erently across di¤erent states of the world. Those factors can be either the returns on some traded securities, non-traded economy wide sources of uncertainty related to macroeconomic variables, or a combination of the two. The empirical success of such models at explaining the so called CAPM anomalies was initially limited, but researchers have progres-sively entertained a broader and broader set of factors, which has resulted in several success claims. Harvey, Liu and Zhu (2016) contains a comprehensive and up to date list of references, cataloguing 315(!) di¤erent factors.
However, several authors have warned that some of those factors, or more generally linear combinations of them, could be uncorrelated with the vector of asset payo¤s that they are meant to price, which would result in economically meaningless models (see Burnside (2016), Gospodi-nov, Khan and Robotti (2015) and the references therein). Further, those papers forcefully argue that such situations can lead to misleading econometric conclusions.
In this context, the purpose of our paper is to study the estimation of risk prices and the testing of the cross-sectional restrictions imposed by overspeci…ed linear factor pricing models. By overspeci…ed models we mean those with at least one non-zero SDF which is uncorrelated with the excess returns on the vector of test assets. We discuss in detail several examples of this situation, which illustrate two important di¤erences between our work and related studies. First, the presence of uncorrelated risk factors is su¢ cient but not necessary for overspeci…cation. As a result, attempts to …nd out which factors are uncorrelated on an individual basis fail to provide a complete answer. Second, overspeci…cation is necessary but not su¢ cient for the model parameters to be underidenti…ed. Therefore, studying parameter identi…cation by means of rank tests does not provide a full answer either.
Our point of departure from the existing literature is that we do not focus exclusively on the properties of the usual estimators and tests. Instead, we use the econometric framework in Arellano, Hansen and Sentana (2012).1 Thus, we can identify a linear subspace of risk prices
compatible with the cross-sectional asset pricing restrictions, a basis of which we can easily parametrize and e¢ ciently estimate using standard GMM methods.
We follow Peñaranda and Sentana (2015) in using single-step procedures, such as the con-tinuously updated GMM estimator (CU-GMM) of Hansen, Heaton and Yaron (1996), to obtain numerically identical test statistics and risk price estimates for SDF and regression methods,
with uncentred or centred moments and symmetric or asymmetric normalizations. GEL meth-ods such as Empirical Likelihood or Exponentially Tilted also share the numerical invariance properties of CU-GMM. However, given that these methods are often more di¢ cult to compute than two-step estimators, and they may sometimes give rise to multiple local minima, we propose simple, intuitive consistent parameter estimators that can be used as sensible initial values, and which will be e¢ cient for elliptically distributed returns and factors. Interestingly, we can also show that these consistent initial values coincide with the GMM estimators recommended by Hansen and Jagannathan (1997), which use the second moment of returns as weighting matrix. For simplicity of exposition, we initially focus on excess returns, but later on extend our analysis to cover gross returns too. Importantly, we show that single-step GMM procedures yield the same numerical results with both types of payo¤s.
In addition to the usual overidenti…cation test, which is informative about the existence of admissible SDFs, we propose simple tests that can diagnose economically meaningless but empirically relevant cases in which the expected values of all SDFs in the identi…ed set are 0, which is equivalent to their being uncorrelated to the test assets. We refer to this situation as complete overspeci…cation, which should not apply to credible empirical models.
In our empirical application, we investigate the potential overspeci…cation of the three-factor extension of the Epstein and Zin (1989) version of the consumption CAPM in Yogo (2006) us-ing quarterly data from the usual Fama and French cross-section of excess returns on size and book-to-market sorted portfolios. Aside from its undisputable in‡uence on the subsequent liter-ature, an important characteristic of this model is that his chosen risk factors were theoretically motivated and not the result of either an extensive search or a reverse engineering process. Nev-ertheless, the results we obtain with our novel inference procedures indicate that the admissible SDFs in the linearized version of this model lie on a two-dimensional subspace, so there is lack of identi…cation. In addition, we cannot reject the null hypothesis that all those SDFs have zero means, which is tantamount to complete overspeci…cation. Importantly, our simulations show that these empirical …ndings are not due to lack of power. On the contrary, if anything, our proposed tests tend to overreject for the sample size of this data set.
Our conclusions about the Yogo (2006) model sharpen the analysis in Lewellen, Nagel and Shanken (2010). These authors show that the strong factor covariance structure in the size and book-to-market portfolios implies that it should be possible to …nd models with macroeconomic factors that can price those test assets. In fact, we …nd not only one (up to scale) admissible SDF using Yogo’s (2006) risk factors, but an entire two-dimensional subspace of SDFs which can price those assets, a situation that standard GMM asymptotic theory cannot cope with.
The rest of the paper is organized as follows. Section 2 introduces linear factor pricing models, and precisely characterizes their potential overspeci…cation. Next, we present our econometric methodology in section 3. Then, we empirically analyze in detail the aforementioned asset pricing model in section 4 and report our simulation evidence in section 5. Finally, we summarize our conclusions and discuss some avenues for further research in section 6. Proofs of formal results and a detailed description of all the possible situations that may arise in models with up to three factors are relegated to appendix A, while appendix B contains the Monte Carlo design.
2
Overspeci…ed Asset Pricing Models
2.1 Stochastic discount factors and moment conditions
Let r be a given n 1 vector of excess returns, whose means E(r) we assume are not all equal to zero. Standard arguments such as lack of arbitrage opportunities or the …rst order conditions of a representative investor imply that
E(mr) = 0
for some random variable m called SDF, which discounts uncertain payo¤s in such a way that their expected discounted value equals their cost.
The standard approach in empirical …nance is to model the SDF as an a¢ ne transformation of some k < n observable risk factors f , even though this ignores that m must be positive with probability 1 to avoid arbitrage opportunities, which would require non-linear speci…cations for m (see Hansen and Jagannathan (1991)). In particular, researchers typically express the pricing equation as
E[(a + b0f)r] = 0 (1)
for some coe¢ cients (a; b), which we can refer to as the intercept and slopes of the a¢ ne SDF m = a + b0f.
We can also estimate the SDF mean c = E(m) by adding the moment condition
E(a + b0f c) = 0; (2)
which exactly identi…es c for given (a; b). A non-trivial advantage of this approach is that (1) and (2) are linear in (a; b; c).2
2
There are two alternative popular approaches to test asset pricing models. One uses Cov(r; f ) instead of
E(rf0) in explaining the cross-section of risk premia, while the other one relies on the regression of r onto a
constant and f . Both require a higher number of parameters to estimate from a higher number of moments, and for that reason we shall not explicitly consider them in this paper. Nevertheless, it is straightforward to extend the results in Peñaranda and Sentana (2015) to our context, so as to prove that all three approaches provide numerically equivalent tests and prices of risk estimates when one uses single-step GMM procedures.
It is pedagogically convenient for our purposes to think about the restrictions the linear factor pricing model above imposes on the parameters (a; b; c) as we increase the number of assets we consider. For simplicity, we focus on the case of two pricing factors.
When n = 1, there is always a two dimensional linear space of admissible solutions, which can be regarded as the dual set to the combination line of expected excess returns and covariances with the risk factors that can be generated by leveraging r1 up or down.
(Figure 1: One asset)
When n = 2, the two dimensional space generated by each asset will generally be di¤erent, so their intersection will be a straight line.3
(Figure 2: Two assets)
Three assets is the minimum number required to be able to reject the model. The reason is the following. If the asset pricing model does not hold, the three linear subspaces associated to each of the assets will only intersect at the origin. We may then say that there is …nancial markets segmentation, in the sense that there is no single SDF within the model that can price all the assets.
(Figure 3: Three segmented asset markets)
If on the other hand the asset pricing model holds, the intersection will be a linear subspace of positive dimension. This requires that the three assets are coplanar in the space of expected excess returns and covariances with the risk factors, so that they all lie on the security market plane. When this happens, we may say that there is …nancial markets integration.
(Figure 4: Three integrated asset markets)
Therefore, when there exist admissible parameter con…gurations other than the trivial one (a; b; c) = (0; 0; 0), we can at best identify a direction in (a; b; c) space, which leaves both the scale and sign of the SDF undetermined. As forcefully argued by Hillier (1990) for single equation IV models, this suggests that we should concentrate our e¤orts in estimating the identi…ed direction. However, empirical researchers often prefer to estimate points rather than directions, and for that reason they typically focus on some asymmetric scale normalization, such as (1; b=a; c=a). In this regard, note that = b=a can be interpreted as prices of risk
3
Occasionally, though, the two linear subspaces might coincide. This will happen when the two assets are collinear in the space of expected excess returns and covariances with the risk factors. We will revisit this issue when we discuss Figure 8 in section 2.2.
since we may rewrite (1) as E(r) = E(rf0) . Other normalizations, such as (a=c; b=c; 1) or b0b+ c2 = 1 are also popular.
(Figure 5: Normalizations)
Nevertheless, given that any asymmetric normalization is potentially restrictive, we prefer to use invariant estimation methods, such as CU-GMM.
In what follows, we consider models in which the elements of f are either non-traded (or treated as such) or they are portfolios of r. In those cases, the pricing conditions (1) and (2) contain all the relevant information to estimate and test the asset pricing model. Nevertheless, it would be very easy to extend our analysis to explicitly deal with traded factors whose excess returns do not belong to the linear span of r. In that case, we should add moment conditions such as
E[(a + b0f)f ] = 0
to (1) and (2) to complete the asset pricing information that we should consider, as Lewellen, Nagel and Shanken (2010) suggest.
2.2 Admissible SDFs sets
The pricing conditions (1) can be expressed in matrix notation as
[ E(r) E(rf0) ] 0 @ a b 1 A = M = 0; (3)
where M is an n (k + 1) matrix of …rst and second moments of data and a (k + 1) 1 parameter vector.
The highest possible rank of M is its number of columns k + 1 because k < n. In that case, though, the asset pricing model will not hold because the only value of that satis…es (3) will be the trivial solution = 0.
On the other hand, if the rank of M is k then there will be a one-dimensional subspace of ’s that satisfy the pricing conditions (3), in which case the solution is unique up to scale, as we explained in the previous section. Not surprisingly, rank(M) = k coincides with the usual identi…cation condition required for standard GMM inference (see e.g. Hansen (1982) and Newey and McFadden (1994)).
Kan and Zhang (1999) and Burnside (2016) among others have forcefully argued that some empirical asset pricing models e¤ectively rely on factors for which the matrix Cov(r; f ) does not have full column rank. The best known example is a useless factor, which would yield a zero
column in the matrix Cov(r; f ). To understand the implications, consider a two-factor model with Cov(r; f2) = 0, so that the matrix M becomes
[ E(r) E(rf1) E(rf2) ] = [ E(r) E(rf1) E(r)E(f2) ]:
Given the rank failure of this matrix in those circumstances, we can always …nd at least a one-dimensional subspace of SDFs whose parameters satisfy (3). Two di¤erent situations might occur.
First, if E(r) and E(rf1) are linearly independent, then rank(M) = 2, the model parameters
will remain econometric identi…ed, and we can still rely on standard GMM inference.4 However, E(r) and E(rf1) linearly independent together with absence of arbitrage opportunities implies
that the true SDF must depend at least on an additional genuine risk factor di¤erent from f1
and f2. As a result, there can be no admissible SDF a¢ ne in the two risk factors selected by the
empirical researcher with a meaningful economic interpretation that can explain cross-sectional risk premia. Indeed, when Cov(r; f2) = 0 but E(r) 6= 0, the SDF conditions (1) will trivially
hold for any m / [f2 E(f2)] because they will all satisfy E(m) = 0 and Cov(r; m) = 0, which
in turn implies that
E(rm) = E(r)E(m) + Cov(r; m) = 0:
As a result, the admissible SDFs will have b1 = 0 and c = E(m) = 0. Thus, this overspeci…ed
model is econometrically identi…ed but economically unattractive.
(Figure 6: Valid but unattractive model with a useless factor)
Second, if f1were a valid pricing factor, so that E(r) = E(rf1) 1, then rank(M) = 1 because
[ E(r) E(rf1) E(rf2) ] = E(r)[ 1 1= 1 E(f2) ]:
Hence, this overspeci…ed pricing model will be economically meaningful but parametrically un-deridenti…ed.
(Figure 7: Valid and attractive model with a useless factor)
More generally, there will be rank failures in Cov(r; f ) whenever we can …nd a valid asset pricing model with fewer factors even though no column of Cov(r; f ) is zero. This will happen in particular if there is only a true pricing factor but we include two di¤erent noisy proxies for it, so that their di¤erence will be uncorrelated to the vector of excess returns. As in the
4
Nevertheless, some asymmetric normalizations may be incompatible with these con…gurations; see section 4.4 in Peñaranda and Sentana (2015) for further details in the case of a single pricing factor.
previous example, this alternative overspeci…ed two-factor model is economically meaningful but parametrically underidenti…ed even though the two candidate pricing factors are correlated with the returns on the test assets.
As a di¤erent example of the same type of overspeci…ed model, assume that both the CAPM and the (linearized) CCAPM hold, in the sense that excess returns on the market and consump-tion growth can price on their own a cross-secconsump-tion of excess returns, i.e. E(r) = E(rf1) 1 and
E(r) = E(rf2) 2. Then, an SDF that is linear in both factors, as in the (linearized) Epstein-Zin
model, implies that rank[(Cov(r; f )] = 1. As a consequence, the matrix M becomes [ E (r) E (rf1) E (rf2) ] = E(r)( 1 1= 1 1= 2 );
which means that we can …nd a two-dimensional subspace of SDFs whose parameters satisfy M = 0, a basis of which will be given by m1 = 1 1f1 and m2 = 1 2f2, with c1 = 1 E(f1) 1
and c2 = 1 E(f2) 2 both di¤erent from 0 because f1 and f2 are correlated with r.
(Figure 8: Two single factor models)
Finally, there will also be a two-dimensional subspace of SDFs whose parameters satisfy M = 0 when there are two useless factors, i.e. Cov(r; f1) = Cov(r; f2) = 0. Speci…cally, if
[ E(r) E(rf1) E(rf2) ] = E(r)[ 1 E(f1) E(f2) ];
then any SDF which is a linear combination of [f1 E(f1)] and [f2 E(f2)] will work.
(Figure 9: Two useless factors)
The special feature of this completely overspeci…ed case is that c = 0 for all admissible SDFs, so there is not only underidenti…cation but also the absence of any meaningful speci…cation.
3
Econometric methodology
3.1 Set estimation
Given our previous discussion, it is of the utmost importance to use statistical inference tools that can successfully deal with situations in which rank(M) k. Following Arellano, Hansen and Sentana (2012), we begin by specifying the dimension of the subspace of solutions to the pricing conditions (3), which we denote d, so that rank(M) = (k + 1) d. Given that we maintain the hypothesis that E(r) 6= 0, we could in principle consider ranks for M as low as 1 or, equivalently, any positive integer d up to a maximum value of k.
As we mentioned before, when d = 1 we can rely on standard GMM to estimate a unique (up to normalization) and use its associated J test to assess the validity of the asset pricing restrictions. However, when d 2, we will have a multidimensional subspace of admissible SDFs even after …xing their scale. Nevertheless, we can e¢ ciently estimate a basis of that subspace by replicating d times the moment conditions (3) as follows:
[ E(r) E(rf0) ] 1 = 0; [ E(r) E(rf0) ] 2 = 0; .. . [ E(r) E(rf0) ] d= 0; 9 > > > > > > > = > > > > > > > ; (4)
and imposing enough normalizations on ( 1; 2; :::; d) to ensure the point identi…cation of a
basis of the null space of M.
In this setting, the familiar J test from the work of Sargan (1958) and Hansen (1982) for overidenti…cation of the augmented model becomes a test for “underidenti…cation”of the original model. The rationale is as follows: if we can identify a linear subspace of risk prices without statistical rejection, then the original asset pricing model is not well identi…ed. In contrast, a statistical rejection provides evidence that the prices of risk in the original model are indeed point identi…ed, unless of course the familiar J test continues to reject its overidentifying restrictions. We can also add moment conditions to estimate (c1; c2; : : : ; cd), which characterize the
ex-pected values of the basis SDF’s. Speci…cally, we can combine (4) with the moment conditions [ 1 E(f0) ] 1 c1 = 0; [ 1 E(f0) ] 2 c2 = 0; .. . [ 1 E(f0) ] d cd= 0; 9 > > > > > > > = > > > > > > > ; (5)
which are exactly identi…ed for given values of ( 1; 2; :::; d).
3.2 Normalizations and starting values
In the presentation of our empirical results, we will use the popular SDF normalization discussed in section 2.1, which …xes the …rst element of i to 1, thereby de…ning the prices of
risk as i = bi=ai. Additionally, we need to impose enough zero restrictions on the prices of
risk to achieve identi…cation.5 Once again, though, the advantage of CU-GMM and other GEL estimators is that our inferences will be numerically invariant to the chosen normalization.
5Alternatively, we could make a d d block of (a permutation of) the matrix (
1; 2; :::; d) equal to the
Nevertheless, one drawback of these single-step methods is that they involve a non-linear optimization procedure even though the moment conditions are linear in parameters, which may result in multiple local minima. For that reason, we propose to use as starting value a computationally simple intuitive estimator that is always consistent, but which would become e¢ cient when the returns and factors are i:i:d: elliptical. This family of distributions includes the multivariate normal and Student t distributions as special cases, which are often assumed in theoretical and empirical …nance.
Let us de…ne (f1; f2; :::; fd) as the vectors of factors that enter each one of the SDFs in (4)
after imposing the necessary restrictions that guarantee the point identi…cation of the basis of risk prices ( 1; 2; :::; d), so that the corresponding Jacobian matrices E(rf0i) have full rank. As
a result, we can re-write (4) as
E 2 6 6 6 6 6 6 4 (1 f1 10 )r (1 f2 20 )r .. . (1 fd d0 )r 3 7 7 7 7 7 7 5 = 0; (6) and (5) as E(1 fi i0 ci) = 0; i = 1; 2; ::; d: (7)
Let rtand ftdenote the values of the excess returns on the n assets and the k factors at time
t. We can then prove that
Proposition 1 If (rt; ft) is an i.i.d. elliptical random vector with bounded fourth moments such
that (6) holds, then:
a) The most e¢ cient GMM estimator of i (i = 1; : : : ; d) from the system (6) will be given by
iT = T X t=1 ~ r+it~r+0it ! 1 T X t=1 ~r+it; (8)
where ~r+it are the relevant elements of the sample factor mimicking portfolios
~r+t = T X s=1 fsr0s ! T X s=1 rsr0s ! 1 rt: (9)
b) When we combine the moment conditions (6) with (7), the most e¢ cient GMM estimator of each i is the same as in a), and the most e¢ cient GMM estimator each ci is the sample mean
of the corresponding SDF.
Intuitively, Proposition 1 states that the optimal GMM estimator in an elliptical setting is such that it prices without error the factor mimicking portfolios in any given sample.
Although the elliptical family is rather broad (see Fang, Kotz and Ng (1990)), it is important to stress that (8) will remain consistent under correct speci…cation even if the assumptions of serial independence and a multivariate elliptical distribution do not hold in practice.
In addition, we can provide a rather di¤erent justi…cation for (8). Speci…cally, we can prove that iT in (8) coincides with the GMM estimator that we would obtain if we used as weighting
matrix the second moment of the vector of excess returns r. In other words, iT minimizes the
sample counterpart to the Hansen and Jagannathan (1997) distance E 1 fi i0 r 0 E rr0 1E 1 fi i0 r
irrespective of the distribution of returns and the validity of the asset pricing model. The reason is that the f.o.c. of this minimization is
E fir0 E rr0 1
E 1 fi i0 r = 0;
which is equivalent to the exact pricing of the factor mimicking portfolios in Proposition 1.
3.3 Testing restrictions on admissible SDF sets
As we have just seen, our inference framework allows us to estimate the set of SDFs that is compatible with the pricing conditions (1). But we can also use it to test if the elements of this set satisfy some relevant restrictions.
A particularly important null hypothesis that empirical researchers would like to …nd evi-dence against is that all SDFs compatible with the data have zero means, a situation we have termed “complete overspeci…cation”. In that case, there will be no element in the admissible SDF set that explains the cross-section of expected returns from a meaningful economic perspec-tive, as we illustrated in section 2.2 for d = 1 and d = 2 in Figures 6 and 9, respectively. Both those …gures show completely overspeci…ed models in which all the SDFs in the correspond-ing admissible set are uncorrelated with the asset payo¤s, which renders them economically uninteresting.
In any given sample, though, the estimated values of the means of the admissible SDFs will not be 0. Given that the SDF means are associated to the parameters (c1; c2; :::; cd) by virtue
of (7), a distance metric (DM) test of H0 : ci = 0 i = 1; : : : ; d will give us a valid test of the null
hypothesis of complete overspeci…cation. As is well known, a DM test simply compares the GMM criterion functions (J statistics) with and without those constraints. We can trivially compute the criterion function without the zero mean constraints from the system (6), or equivalently, from the joint system that also considers the exactly identi…ed moment conditions (7). In turn, we can construct the criterion function that imposes the zero mean constraints on all the SDFs from the system
E 2 4 (1 fi i0 )r 1 fi i0 3 5 = E (1 fi i0 )x = 0; i = 1; 2; ::; d; (10)
where x0 = (r0; 1), which is analogous to (6) for an extended vector of payo¤s that includes a …ctional unit safe payo¤.6
Again, normalization-invariant procedures are crucial to avoid obtaining di¤erent results for di¤erent basis of the admissible SDF set. But given the numerical complications that they may entail, we again propose to use as starting value a computationally simple intuitive estimator that is always consistent, but which would become e¢ cient when the returns and factors are i:i:d: elliptical. In fact, we can prove that the optimal estimator of the prices of risk continues to have the same structure as in Proposition 1 if we de…ne the factor mimicking portfolios over the extended payo¤ space. Speci…cally:
Proposition 2 If (rt; ft) is an i.i.d. elliptical random vector with bounded fourth moments such
that (10) holds, then the most e¢ cient GMM estimator of i (i = 1; : : : ; d) will be given by
_iT = T X t=1 ~ x+it~x+0it ! 1 T X t=1 ~ x+it; (11)
where ~x+it are the relevant elements of the sample factor mimicking portfolios
~ x+it = T X s=1 fsx0s ! T X s=1 xsx0s ! 1 xt: (12)
Another interesting null hypothesis that we may also want to test is whether some partic-ular pricing factor does not appear in any admissible SDF. Formally, the corresponding null hypothesis would be that the entry of b associated to this factor being zero in all the vectors ( 1; 2; :::; d).
(Figure 10: An unpriced second factor)
Again, a DM test based on single-step GMM procedures will be ideally suited for testing this restriction on the space of admissible SDFs.7
3.4 Comparison to the existing literature
Burnside (2016) and Gospodinov, Kan and Robotti (2015) study the identi…cation of the prices of risk of the linear factor pricing model (1) (or its centred version in (A1), with m =
6If there really existed an unconditionally safe asset, an SDF that satis…ed E (xm) = 0 would allow for
arbitrage opportunities in the extended payo¤ space. Although no such an asset exists in real life, the fact that all the SDFs in the admissible set satisfy those moment conditions signals the problematic economic interpretation of a completely overspeci…ed model.
7
Given that the moment conditions (5) and (6) are linear in parameters and the restrictions to test are homogenous, the results in Newey and West (1987b) imply that the Wald, Lagrange Multiplier and DM tests would be numerically identical for two-step GMM methods that shared the same weighting matrices. More generally, though, DM tests might be more reliable than Wald tests in non-standard situations with potential identi…cation failures (see Dufour (1997) for closely related results in a likelihood context).
c + b0(f ), where = E(f ) is the vector of risk factor means) by applying the tests proposed by Cragg and Donald (1997) and Kleibergen and Paap (2006) to assess the rank of E(rf0) (or Cov(r; f )), which coincide with the expected Jacobian matrices of the GMM conditions (3) (or their centred counterparts).
As we mentioned in the introduction, though, overspeci…cation is a necessary but not su¢ -cient condition for underidenti…cation. In that regard, we can prove the following result: Proposition 3 The CU version of the overidenti…cation test of the original SDF moment con-ditions (4) and (5) after imposing the d restrictions c1 = : : : = cd = 0 numerically coincides
with the CU version of the test of the null hypothesis rank[Cov(r; f )] = k d.
As a result, the DM test of c1 = ::: = cd = 0 we introduced in the previous section can be
interpreted as a test of the null hypothesis that rank[Cov(r; f )] = k d under the maintained hypothesis that rank(M) = (k + 1) d. In other words, this DM test implicitly checks whether the di¤erence in the rank of those two matrices is one, or equivalently, whether E(r) cannot be spanned by Cov(r; f ), in which case the only admissible SDFs would be those economically meaningless random variables that exploit the rank failure in Cov(r; f ) in setting to zero the pricing conditions (1).
In contrast, the test of the rank of Cov(r; f ) in Proposition 3 is often uninformative about the existence of economically meaningful SDFs precisely because it does not maintain any hypothesis on the rank of M.8 For example, both in the model with a valid factor and a useless one depicted
in Figure 7, and in the double single factor model described in Figure 8, the matrix Cov(r; f ) has rank 1 instead of 2 while M has rank 1 instead of 3. As a result, E(r) belongs to the span of Cov(r; f ), which con…rms that in those two examples there exist economically meaningful SDFs that correctly price r.
On the other hand, Figure 6 shows another possible situation with rank[Cov(r; f )] = 1 instead of 2 in which E(r) cannot be spanned by Cov(r; f ) because the rank of M is 2, so that the only admissible SDFs must be uncorrelated to the vector of excess returns.
The other main di¤erence with those papers is that they focus on the implications of those rank failures for standard GMM procedures, which assume point identi…cation, while we propose alternative inference procedures that explicitly handle set identi…cation.
3.5 Gross returns
Let R denote a vector of gross returns on N = n + 1 assets. Without loss of generality, we can understand the vector of excess returns r that we have used so far as the di¤erence between
8
The only exception is the extreme case of Cov(r; f ) = 0, which necessarily means rank(M) = 1 when E(r) 6= 0, making it impossible to …nd meaningful SDFs that can explain E(r), as Figure 9 illustrates.
the gross returns of the last n assets and the …rst one, R say. In practice, this reference asset could be the real return on US T-bills, whose payo¤s are not constant. The relevant pricing equation for R becomes:
E[R(a + b0f)] = `;
where ` is a vector of N ones. Without loss of generality, we can re-write these moment conditions as the combination of (1) with:
E R(a + b0f) = 1: (13)
In addition, we can continue to estimate the SDF mean from the moment condition (2). The addition of the pricing of R in (13) implies that we no longer require an arbitrary normalization of (a; b; c). As Peñaranda and Sentana (2015) prove in their Appendix A, though, the empirical evidence obtained by single-step methods applied to R is consistent with the analogous evidence obtained from r alone. In particular, the overidenti…cation restriction test for the joint system (1) and (13) is numerically identical to the one for (1) alone, and the normalized estimate of obtained from the moment conditions for excess returns coincides with the ratio of the estimates of b to a obtained using all the assets. Intuitively, the addition of gross returns allows us to pin down a and the mean of the SDF, c, but otherwise, it simply re-scales this variable.
More formally, we can re-express the SDF as a(1 f0 ) and re-write the moment conditions (1) and (13) as E[(1 f0 )r] = 0 and E[a(1 f0 )R] = 1, respectively. This last equation does not contain any additional information about the prices of risk , it only pins down a, or equivalently c.
Finally, the same comments apply to those situations with d > 1. The only di¤erence is that they involve several SDFs of the form ai(1 fi i0 ) for i = 1; : : : ; d. But since we add one moment
and one parameter for each dimension, the equivalence between the results for excess and gross returns we have just discussed for d = 1 continues to hold for any d.
4
Empirical Application
4.1 Original results
To illustrate the practical relevance of our proposed methods, in this section we analyze in detail the linear version of the three-factor model in Yogo (2006) in order to assess the potential overspeci…cation of this popular empirical model.
As is well known, his theoretical model extends the CCAPM by assuming recursive prefer-ences over a consumption bundle of nondurable and durable goods.9 Therefore, in the linearized
9
version of his model, the SDF will depend on three factors: the market return, and the con-sumption growth of nondurables and durables, so that we can write it as:
m = a (1 1f1 2f2 3f3) :
In practice, the log-growth rate of US real per capita consumption of nondurables and services and durables are identi…ed with f2 and f3, respectively. In turn, the return on wealth - proxied
by the (log) return on the value-weighted U.S. stock market measured in real terms - is associated with f1.
We initially evaluate this model with the original data, which corresponds to quarterly excess returns on the Fama-French cross-section of 25 size and book-to-market sorted portfolios from 1951 to 2001 (see Fama and French (1993) for further details).10 In addition to the insightful nature of Yogo’s (2006) theoretically motivated SDF speci…cation, his results became very in‡uential because not only did he fail to reject the asset pricing restrictions but also because he succeeded in aligning the risk premia in the data with the risk premia generated by his model.
(Figure 11: Risk premia from 2S-GMM)
Nevertheless, the theoretical results in Burnside (2016) and Gospodinov, Kan and Robotti (2015) indicate that a high cross-sectional R2may spuriously arise in models with useless factors. As an aside, we …nd that the results in Figure 11 depend on the estimation method (2-step GMM) and the imposition of some restrictions on the prices of risk.11 Speci…cally, if we use instead iterated GMM starting from the 2-step estimates, we encounter a cycle with four di¤erent solutions.
(Figure 12: Risk premia from IT-GMM)
Convergence does not improve if we free up the price of risk coe¢ cients: iterated GMM enters yet another cycle of three di¤erent solutions.
(Figure 13: Risk premia from IT-GMM, free coe¢ cients) These discrepancies highlight the advantages of single step methods.
necessary to look at di¤erent consumption subaggregates to successfully explain asset risk premia.Yogo’s (2006) empirical model goes one important step further by combining their ideas with those in Epstein and Zin (1989).
1 0
Note that although the market return is a traded factor, we do not add its pricing condition to (1) because it can e¤ectively be generated as a portfolio of the cross-section of excess returns that we want to price.
1 1
In this section, we follow Yogo (2006) in using the moment conditions of the centred SDF approach mentioned in footnote 2, which are given by equation (A1) in the proof of Proposition 2 with the normalization c = 1. As Peñaranda and Sentana (2015) argue, this matters for two step and iterated GMM, but not for CU-GMM.
4.2 Overspeci…cation
The sensitivity of the empirical results to the estimation method might be a sign of over-speci…cation. For that reason, we apply our methodology to the same data. Speci…cally, we use the moment conditions (6) with d = 1, 2 and 3 to test for one, two and three-dimensional linear subsets of valid SDFs, respectively. In all cases, we augment those moment conditions with the exactly identi…ed moment conditions (7) to obtain the associated SDF means. As we mentioned in section 3.3, we can then assess whether the model is completely overspeci…ed by testing the joint signi…cance of those means.
We estimate the di¤erent subspaces for risk prices and SDFs using single-step GMM methods choosing those normalizations which are arguably easiest to interpret in each context. In the case of d = 2, in particular, we present the results for the simple normalization of the prices of risk given by ( 1; 2; 0) and ( 1; 0; 3).12 Since the …rst factor is the market, we can interpret those
two SDFs as two variants of the linearized Epstein and Zin (1989) model, one with nondurable consumption and another with durable consumption. In contrast, in the case of d = 3 we present the results for the simple normalization ( 1; 0; 0), (0; 2; 0) and (0; 0; 3), which e¤ectively imposes
that each factor can separately explain risk premia.
Table 1 shows the results of our overspeci…cation analysis of the model. This table displays estimates of the SDF parameters and associated J tests. The criterion function is also reported under the restriction of zero SDF means, which is equivalent to a rank test for Cov(r; f ) from Proposition 3. The p-values of the di¤erent J tests are shown in parenthesis.
We complement the J tests with signi…cance tests for the SDF prices of risk. In particular, to the right of the point estimates we report in parenthesis the p-value of the DM test of the null hypothesis of a zero parameter value. The …rst, second and third blocks of columns refer to SDF sets of dimension 1, 2 and 3, respectively. All the reported results correspond to a weighting matrix a la Newey and West (1987a) with one lag, but we obtain qualitatively similar conclusions when we use a VARHAC procedure also with one lag.13
(Table 1: Empirical evaluation of the model 1951-2001)
The results for the one-dimensional set entirely agree with the results in Yogo (2006), who …nds that (i) the J test of two-step GMM does not reject his model for these 25 size- and value-sorted portfolios and (ii) durable consumption provides the only non-zero price of risk. In
1 2
This normalization is identi…ed as long as 2 6= 0 and 3 6= 0. In that regard, Table 1 below shows that the
DM tests that we proposed in section 3.3 reject both 2= 0and 3= 0:
1 3
Den Haan and Levin’s (1997) VARHAC procedure assumes that the moment conditions have a …nite VAR representation, which they exploit to estimate the required long run covariance matrix.
this regard, the usual overidenti…cation test reported in the …rst column of Table 1 does not reject the null hypothesis that there exists an SDF a¢ ne in the three factors that can price the cross-section of securities (p-value=53:7%).
However, the validity of the asymptotic distribution of this J test crucially depends on the model parameters being point identi…ed. For that reason, we also report the overidenti…cation test for d = 2. As explained before, this test assesses whether there is a linear subspace of dimension 2 of admissible SDFs that can price the cross section of risk premia. We obtain a p-value of 13:4%, which suggests that a model with the market together with durable and nondurable consumption as risk factors might be overspeci…ed. In contrast, the overidenti…cation test corresponding to d = 3 is rejected.
Therefore, we …nd evidence that the admissible SDFs of this model lie on a two-dimensional subspace. In addition, the DM test of the null hypothesis that all the admissible SDFs have zero means when d = 2 has a p-value of 49:4%. This suggests that the seeming pricing ability of this set of SDFs simply exploits the lack of correlation of its elements with r. In other words, the vector of risk premia does not appear to lie in the span of the covariance matrix of the excess returns and the factors, which suggests the model is completely overspeci…ed.
Our results are in line with Burnside (2016), who …nds that the matrix Cov(r; f ) for this combination of test assets and risk factors has rank 1 only. His evidence implies that there are SDFs that price the test assets in an economic unattractive manner. Our results con…rm that those SDFs seem to be the only admissible ones that a linearized extension of Epstein and Zin (1989) CCAPM combining durable and nondurable consumption can generate.
4.3 Robustness exercises
One potential concern with our methods is that the number of moments involved may be too large relative to the sample size. For that reason, we assess the reliability of the empirical results in Table 1 in two di¤erent ways: using a sample with a longer time span, and also with a smaller cross-section of test assets. In the …rst case, we use the same data as Burnside (2016), whose sample period is 1949-2012, while in the second one we make use of the 6 size-and value-sorted Fama-French portfolios over the same time span.
Table 2 shows that our main …ndings are robust to these changes.14 Speci…cally, we continue to …nd that the admissible SDFs lie on a two-dimensional subspace, and that the entire set of admissible SDFs has zero means.
1 4
We follow Burnside (2016) in using real excess returns, while Yogo (2006) used nominal excess returns. Given that the e¤ect of in‡ation is second order for excess returns, the choice between nominal and real returns is inconsequential for our results.
(Table 2: Empirical evaluation of the model 1949-2012)
5
Monte Carlo Evidence
In this section, we assess the …nite sample size and power properties of the testing procedures discussed above by means of several extensive Monte Carlo exercises. The exact design of our experiments is described in Appendix B. Given that the number of mean, variance and correlation parameters for returns and factors is large, we have simpli…ed the data generating process (DGP) as much as possible without losing generality, so that in the end we only had to select a handful of parameters whose interpretation is very simple.
We use n = 6 and T = 200. This number of test assets coincides with the dimension of one of the Fama-French cross-sections in the previous section, while the sample size represents …fty years of quarterly data. Further, we also run simulations with T = 600, which corresponds to …fty years of monthly data. In all instances, we simulate 10; 000 samples for each design.
5.1 Numerical details
The main practical di¢ culty is that we have to rely on numerical optimization methods to maximize the non-linear CU-GMM criterion function even though the moment conditions are linear in the parameters. For that reason, we compute the criterion by means of the auxiliary OLS regressions described in appendix B of Peñaranda and Sentana (2012). We achieve further gains in numerical reliability by using the consistent estimators in Propositions 1 and 2 as starting values.
Given that single-step methods are invariant to di¤erent parametrizations of the SDF, we use the uncentered version in (6) because it is the most parsimonious in terms of parameters. Nevertheless, one could exploit the numerical equivalence of the di¤erent approaches mentioned in footnote 2, as well as the di¤erent normalizations, to check that a global minimum has been reached.
In view of the exactly identi…ed nature of the moment conditions (5), further speed gains can be achieved by minimizing the original moment conditions (6) with respect to 1; : : : ; d
only. Once this is done, the joint criterion function can be minimized with respect to c1; : : : ; cd
only, keeping 1; : : : ; d…xed at their CUEs and using the sample means of the estimated SDF
basis as consistent starting values.
5.2 Two-dimensional set of admissible SDFs
Table 3 displays the rejection rates of the J and DM tests when there is a two-dimension set of admissible SDFs. In our two factor setting, this means that any of the factors can price
the cross-section of returns on its own. Our standard asymptotic theory implies that we expect rejection rates close to size for the J test for d = 2. In contrast, the usual J test for d = 1 should under-reject because of its generic lack of identi…cation. The only exception arises when c = 0, in which case there will be a unique linear combination of the factors that yields an admissible SDF with zero mean, even though the two SDFs that we use in this design have nonzero means. Thus, the J test for d = 1 that imposes a zero SDF mean should yield rejection rates close to size too.
Panel A reports the rejection rates when most SDFs in the admissible set have nonzero means, while Panel B shows the corresponding …gures when the asset pricing model is completely overspeci…ed. To achieve this, we use two factors that are uncorrelated with the cross-section of returns as the DGP of Panel B. In each panel, we report the rates for 6 tests: the J tests for d = 2 and d = 1, their variants restricted to have zero SDF means, and the corresponding DM tests.
(Table 3: Rejection rates for a two-dimensional set of admissible SDFs)
The …rst result we can see in Panel A of Table 3 is that the J test for d = 2 performs well, showing only a slight overrejection under the null, and considerable power against c = 0. As expected, the J test for d = 1 massively under-rejects when we do not impose the restriction that c = 0; while it has rejection rates close to size if we do.
On the other hand, Panel B of Table 3 con…rms that the J test for d = 2 underrejects, the restricted J test performs well, with only a slight overrejection, and the corresponding DM test overrejects. This last overrejection indicates that the fact that this DM test does not reject in our empirical application is not due to lack of power. In that regard, Table B1 in the Appendix shows that this DM test no longer shows any noticeable size distortions for T = 600.
5.3 One-dimensional set of admissible SDFs
Table 4 displays the rejection rates of the J and DM tests when the true model contains only one (up to scale) admissible SDF. In that case, we expect that the J test for d = 1 yields rejection rates close to size, while the J test for d = 2 should now show substantial power.
Once again, Panel A contains the rejection rates when the SDF has a nonzero mean, while Panel B reports the corresponding …gures when the model is overspeci…ed. To achieve this, we impose that one the factors is uncorrelated with the cross-section of returns as the DGP of Panel B.
As expected, Panel A of Table 4 con…rms that the J test for d = 1 performs well while the J test for d = 2 has indeed power. Therefore, our …nding an overspeci…ed model in the empirical application cannot be due to lack of power of this second test.
In Panel B of Table 4, the J test for d = 2 shows considerable power. Further, the J test for d = 1 underrejects, the restricted J test performs well, and the corresponding DM test overrejects. As in the previous section, Table B2 in the Appendix shows that this DM test no longer shows any noticeable size distortions for T = 600.
Finally, we also simulated a design where the admissible set of SDFs is empty, as described in Appendix B. In this case, all the tests that we study should reject their respective null hypotheses. Our results, which are available upon request, con…rm the power of our proposed procedures such a design.
6
Conclusions
We study the estimation of risk prices and the testing of the cross-sectional restrictions imposed by overspeci…ed linear factor pricing models in which there is at least one non-zero SDF which is uncorrelated with the excess returns of the test assets chosen by the researcher. We provide several examples of this situation, which is necessary but not su¢ cient for the model parameters to be underidenti…ed. In addition, we also emphasize the distinction between a model with uncorrelated pricing factors, which is necessarily overidenti…ed, from a model with uncorrelated SDFs.
Unlike previous studies, which focus on the non-standard asymptotic properties of the usual estimators and tests, our methods directly estimate the linear subspaces of prices of risk and associated SDFs compatible with the pricing restrictions of the model, which we can easily express in terms of linear moment conditions and e¢ ciently estimate using standard GMM methods. In this regard, a non-trivial advantage of our unusual procedures is that they have standard asymptotic distributions.
We use single-step GMM procedures, and in particular continuously updated GMM, to obtain identical test statistics and risk price estimates for SDF and regression methods, with uncentered or centred moments and symmetric or asymmetric normalizations. Another non-trivial advantage of these methods is that they yield exactly the same conclusions for excess returns and gross returns.
We also propose simple tests to detect economically unattractive but empirically relevant situations in which the expected values of all SDFs in the identi…ed set are 0, which is equivalent to their being uncorrelated to the test assets. In our opinion, researchers could convince readers
that their results are meaningful by systematically reporting that they reject the restrictions implicit in these completely overspeci…ed models.
In our empirical application, we investigate the potential overspeci…cation of the three-factor extension of the Epstein and Zin (1989) version of the consumption CAPM model in Yogo (2006), which combines two macroeconomic factors: non-durable and durable consumption, and a stock market factor.
We evaluate the linearized version of this model with the original data, which corresponds to excess returns on the Fama-French cross-section of 25 size and book-to-market sorted portfolios from 1951 to 2001. Our results indicate that the admissible SDFs lie on a two-dimensional subspace. In addition, we cannot reject the null hypothesis that model is completely overspeci-…ed. Importantly, our results hold both when we update the sample and when we consider the Fama-French cross-section of 6 portfolios. In addition, our simulations show that these empirical …ndings are not due to lack of power. On the contrary, if anything, our proposed tests tend to overreject for the sample sizes of our datasets.
Our econometric methodology is positive in nature, in the sense that our main objective has been to complement the diagnostics that researchers typically report in support of their preferred linear factor pricing speci…cation so as to increase the empirical credibility of their results. Nevertheless, it might be interesting to combine our procedures with normative econometric methods that some researchers use to come up with an acceptable speci…cation. Three recent proposals are Harvey, Liu and Zhu (2016), Bryzgalova (2016) and Kozak, Nagel and Santosh (2017). The application of our proposed diagnostics to models that have been selected after an implicit or explicit speci…cation search raises multiple testing issues that we leave for future research.
Another interesting avenue for further research would be to consider bootstrap versions of our tests to improve their …nite sample reliability. Finally, we could also apply our methods to other popular empirical asset pricing models, such as the ones in Jagannathan and Wang (1996) or Lettau and Ludvigson (2001). We are currently pursuing some of these extensions.
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Appendices
A
Proofs and special cases
A.1 Proofs
Proposition 1
We develop most of the proof for the case d = 2 to simplify the expressions, but explain the extension to d > 2 at the end.
a) When d = 2, the moment conditions (6) become
E (m r) = E 0 @ m1r m2r 1 A = E 0 @ (1 f1 10 ) r (1 f2 20 ) r 1 A = 0:
We know from Hansen (1982) that the optimal moments correspond to the linear combina-tions D0S 11 T T X t=1 0 @ m1trt m2trt 1 A ;
where D is the expected Jacobian and S the corresponding long-run variance
S= avar 2 4p1 T T X t=1 0 @ m1trt m2trt 1 A 3 5 : In this setting, the expected Jacobian trivially is
D= 0 @ D1 0 0 D2 1 A ; Di= E rf0i :
Since we assume that the chosen normalization ( 1; 2) is identi…ed, D has full column rank,
which in turn implies that both D1 and D2 must have full column rank too.
When (rt; ft) is an i.i.d. elliptical random vector with bounded fourth moments, we can
tediously show that the long-run covariance matrix of the in‡uence functions will be S= A E rr0 B E (r) E (r)0;
A = (1 + ) V (m) + E (m) E (m)0; B = V (m) + 2 (1 ) E (m) E (m)0; where is the coe¢ cient of multivariate excess kurtosis (see Fang, Kotz and Ng (1990)).
To relate the optimal moments to the factor mimicking portfolios r+i = Cir; Ci = E rf0i 0E 1 rr0 ;
it is convenient to de…ne the matrix C0 = 0 @ C01 0 0 C02 1 A ;
on the basis of which we can compute SC0 = A E rr0 B E (r) E (r)0 0 @ C01 0 0 C02 1 A = 0 @ A11E rf01 A12E rf02 A12E rf01 A22E rf02 1 A 0 @ B11E (r) E (r)0C01 B12E (r) E (r)0C02 B12E (r) E (r)0C01 B22E (r) E (r)0C02 1 A :
Given that the existence of two valid SDFs implies that E(r) = E(rf01) 1 = E(rf02) 2, we
can write these matrices as
SC0 = 0 @ A11E rf01 A12E rf02 A12E rf01 A22E rf02 1 A 0 @ B11E rf01 1 01G1 B12E rf02 2 02G2 B12E rf01 1 01G1 B22E rf02 2 02G2 1 A ; Gi= E rf0i 0 E 1 rr0 E rf0i :
In addition, let us de…ne the matrices Qi such that E(rf01) = E(rf02)Q1 and E(rf02) =
E(rf01)Q2, which are related by Q2 = Q11. The existence of these matrices is guaranteed by
the lack of full column rank of E(rf0) together with the full column rank of E(rf01) and E(rf02). Thus, we can write
SC0 = DQ; Q = 0 @ A11I1 B11 1 01G1 Q2 A12I1 B12 2 02G2 Q1 A12I2 B12 1 01G1 A22I2 B22 2 02G2 1 A :
The assumption that D0S 1 has full row rank guarantees that the same is true for C, so that Q will be invertible. Therefore, we have found that
D0S 1= Q0 1C:
In other words, the rows of D0S 1 are spanned by the rows of C, which con…rms that the factor mimicking portfolios span the optimal instrumental variables.
As a result, the optimal moments can be expressed as 0 @ C1 0 0 C2 1 A 1 T T X t=1 0 @ m1trt m2trt 1 A = 1 T T X t=1 0 @ r + 1tm1t r+2tm2t 1 A = 0;
which proves that the optimal estimator of each vector of risk prices simply uses the correspond-ing factor mimickcorrespond-ing portfolios. This estimator is infeasible because we do not know Ci, but
under standard regularity conditions we can replace r+it by its sample counterpart in (9) without a¤ecting the asymptotic distribution.
b) When d = 2, the joint system of moments (6) and (7) E (h) = E 0 @ m r m c 1 A ; is composed by E (m r) = E 0 @ m1r m2r 1 A = E 0 @ (1 f1 10 ) r (1 f2 20 ) r 1 A = 0; E (m c) = E 0 @ m1 c1 m2 c2 1 A = E 0 @ 1 f1 10 c1 1 f2 20 c1 1 A = 0; with the parameters being
= 0 @ c 1 A ; = 0 @ 1 2 1 A ; c= 0 @ c1 c2 1 A : The optimal moments correspond to the linear combinations
D0S 1T1
T
X
t=1
ht;
where D is the expected Jacobian and S the corresponding long-run variance S = avar " 1 p T T X t=1 ht # : In this setting, the expected Jacobian can be decomposed as
D = 0 @ D 0 D I2 1 A ;
where D contains the Jacobian of m c with respect to , and I2 is the identity matrix of order
2. The long-run variance for i.i.d. returns and factors can be decomposed as
S = 0 @ S E (mm0 r) E (mm0 r0) V ar (m) 1 A :
Once again, we can exploit the structure of the optimal moments to show that the optimal estimator of satis…es the moment conditions
D0S 11 T T X t=1 (mt rt) = 0:
Hence, the optimal estimator of c will satisfy the moment conditions 1 T T X t=1 (mt c) E mm0 r0 S 1 1 T T X t=1 (mt rt) = 0:
Obviously, as the additional moments E (m c) = 0 are exactly identi…ed, the moment conditions that de…ne the optimal estimator of coincide with the conditions in point a), and consequently the same estimator is obtained. The optimal estimator of c is equal to
1 T T X t=1 mt E mm0 r0 S 1 1 T T X t=1 (mt rt) ;
with mt evaluated at the optimal estimator of .
When (rt; ft) is an i.i.d. elliptical random vector with bounded fourth moments, we can show
that
E mm0 r0 = C E (r)0; C = V ar (m) E (m) E (m)0: There are two valid SDFs: E (r) = E rf01 1= E rf02 2. Thus, we can write
E mm0 r0 = 0 @ C11E (r)0 C12E (r)0 C12E (r)0 C22E (r)0 1 A = 0 @ C11 01E rf01 0 C12 02E rf02 0 C12 01E rf01 0 C22 02E rf02 0 1 A : Let us focus on the optimal estimator of c1. We can express it as
1 T T X t=1 m1t C11 01 C12 02 0 @ E rf01 0 0 0 E rf02 0 1 A S 11 T T X t=1 (mt rt) = 1 T T X t=1 m1t+ C11 01 C12 02 D0S 11 T T X t=1 (mt rt) ;
where the second term must be zero by de…nition of the optimal estimator of . A similar argument can be applied to the optimal estimator of c2. Hence, we can conclude that
^ c= 1 T T X t=1 mt
will be the optimal estimator of the SDF means in an elliptical setting.
Finally, we can easily extend our proof to d > 2 because the structures of D, S, and C are entirely analogous. Speci…cally, S will continue to be the same function of A and B above, although the dimension of these matrices becomes d instead of 2. In turn, D and C will remain block-diagonal, but with d blocks instead of 2 along the diagonal. Lastly, E (mm0 r0) will continue to be the same function of C above.
Proposition 2
One again, we develop most of the proof for the case d = 2 to simplify the expressions, but explaining the extension to d > 2 at the end.
When d = 2, the moment conditions (10) become E (m x) = E 0 @ m1x m2x 1 A = E 0 @ (1 f1 10 ) x (1 f2 20 ) x 1 A = 0: The optimal moments correspond to the linear combinations
D0S 11 T T X t=1 0 @ m1txt m2txt 1 A ;
where D is the expected Jacobian and S the corresponding long-run variance. In this setting, the expected Jacobian is block-diagonal with blocks E xf0i .
When (rt; ft) is an i.i.d. elliptical random vector with bounded fourth moments, and E (m) =
0, we can use our previous results in the proof of Proposition 1 to show that the long-run covariance matrix of the in‡uence functions will be
S= A E xx0 B E (x) E (x)0; A = (1 + ) E mm0 ; B = E mm0 ; where is the coe¢ cient of multivariate excess kurtosis.
The structure of D and S are similar to the proof of Proposition 1. Therefore, we can follow the same argument to conclude that if we de…ne the factor mimicking portfolios on the extended payo¤ space as
x+i = Cix; Ci = E xf0i 0E 1 xx0 ;
then the sample version of the optimal moments can be expressed as 0 @ C1 0 0 C2 1 A 1 T T X t=1 0 @ m1txt m2txt 1 A = 1 T T X t=1 0 @ x + 1tm1t x+2tm2t 1 A :
This expression proves that the optimal estimator of each vector of risk prices simply uses the corresponding factor mimicking portfolios. Once again, this estimator is infeasible because we do not know Ci, but under standard regularity conditions we can replace x+it by its sample
counterpart in (12) without a¤ecting the asymptotic distribution.
As in the case of Proposition 1, we can easily extend our proof to d > 2 because the structures of D, S, and C is entirely analogous. Speci…cally, S will continue to be the same function of A and B above, although the dimension of these matrices becomes d instead of 2: In turn, D and Cwill remain block-diagonal, but with d blocks instead of 2 along the diagonal.
Proposition 3
The proof is simpler if we express the pricing conditions (1) in terms of central moments, which is numerically inconsequential for CU-GMM. Speci…cally, we can add and subtract b0 from a + b0f and de…ne c = a + b0 as the expected value of the a¢ ne SDF. This allows us to
re-write the pricing conditions as
E 8 < : [c + b0(f )] r f 9 = ;= 0: (A1)
In this way, the unknown parameters become (c; b; ) instead of (a; b), as we have added k extra moments to estimate . Empirical researchers often use the implicit normalization c = 1, but this would be incompatible with the null hypothesis H0 : c = 0 that we want to test. In contrast,
the symmetric normalization b0b+ c2= 1 is perfectly compatible with this null hypothesis. To deal with a d dimensional subspace of admissible SDFs, we need to replicate d times the pricing conditions in (A1). Thus, the centred SDF counterpart to (4) will be based on the moment conditions E 0 B B B B B B @ f rm1 .. . rmd 1 C C C C C C A = 0; mi= ci+ (fi i)0bi; (A2)
where (f1; f2; :::; fd) are the vectors of factors that enter each one of the SDFs after imposing the
necessary restrictions that guarantee the point identi…cation of the basis (b1; b2; :::; bd).
Let us denote by J the CU-GMM value of the overidentifying restrictions test with free (c1; c2; :::; cd) in (A2). Similarly, let us denote by J0 the CU-GMM value of the corresponding
overidentifying restrictions test after imposing c1 = ::: = cd= 0. In this context, it is
straight-forward to see that the overidenti…cation test based on J0 is trivially a rank test on Cov(r; f )
because it is testing the existence of d linear combinations of the columns of this covariance matrix with weights bi that are equal to zero. By the invariance properties of single-step GMM
methods, it is easy to prove that we would obtain the same value for the overidenti…cation test from the moment conditions (4) and (5).
Finally, note that our DM test of the null hypothesis c1= ::: = cd= 0 is based on J0 J .
A.2 Possible cases with one, two and three factors
We describe all the possible cases for models with one, two or three factors under the maintained assumption that E(r) 6= 0. As a result, we only study cases where the rank of M is one or higher.
